A030009 Euler transform of primes.

1, 2, 6, 15, 37, 85, 192, 414, 879, 1816, 3694, 7362, 14480, 28037, 53644, 101379, 189587, 350874, 643431, 1169388, 2108045, 3770430, 6694894, 11804968, 20679720, 35999794, 62298755, 107198541, 183462856, 312357002, 529173060, 892216829, 1497454396, 2502190992

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

Formula

G.f.: Product_{n>=1} (1-x^n)^(-prime(n)).

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*ithprime(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  #  Alois P. Heinz, Sep 06 2008

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*Prime[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 16 2014, after Alois P. Heinz *)

Prog

(PARI) a(n)=if(n<0, 0, polcoeff(prod(i=1, n, (1-x^i)^-prime(i), 1+x*O(x^n)), n))
(SageMath) # uses[EulerTransform from A166861]
b = EulerTransform(lambda n: nth_prime(n))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

Crossrefs

Cf. A007441.

Sequence in context: A017923 A238830 A018018 * A061261 A335903 A291414

Adjacent sequences: A030006 A030007 A030008 * A030010 A030011 A030012

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved